Problem: All of the 5th grade teachers and students from Oak went on a field trip to an archaeology museum. Tickets were $$7.00$ each for teachers and $$4.50$ each for students, and the group paid $$32.00$ in total. The next month, the same group visited a science museum where the tickets cost $$14.00$ each for teachers and $$12.50$ each for students, and the group paid $$78.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+4.5y = 32}$ ${14x+12.5y = 78}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-14x-9y = -64}$ ${14x+12.5y = 78}$ Add the top and bottom equations together. $ 3.5y = 14 $ $ y = \dfrac{14}{3.5}$ ${y = 4}$ Now that you know ${y = 4}$ , plug it back into $ {7x+4.5y = 32}$ to find $x$ ${7x + 4.5}{(4)}{= 32}$ $7x+18 = 32$ $7x = 14$ $x = \dfrac{14}{7}$ ${x = 2}$ You can also plug ${y = 4}$ into $ {14x+12.5y = 78}$ and get the same answer for $x$ ${14x + 12.5}{(4)}{= 78}$ ${x = 2}$ There were $2$ teachers and $4$ students on the field trips.